For the same reasons as for measures, one needs a way of comparing two mm-spaces to quantify their similarity.
Sturm's distance [1]} combines ideas of the Gromov-Hausdorff distance [2]}, [3]}, [4]} with OT.
It consists in mapping two mm-spaces \(=(, d_,)\)  and \(=(, d_,)\)  isometrically into a third space \((,d_)\) , and performing OT on that third space.
It reads
\(d_{Sturm}(,)\triangleq \inf _{(,d_), \psi _,\psi _} _{d_}((\psi _)_\sharp , (\psi _)_\sharp ),\) 
